Core partitions into distinct parts and an analog of Euler’s theorem

Straub, Armin

doi:10.1016/j.ejc.2016.04.002

Cited by 39 publications

(64 citation statements)

References 17 publications

(17 reference statements)

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“…Summing shows that the total number of (s, s + 1)-cores with any number of distinct parts is the Fibonacci number F ib s+1 . This fact was originally conjectured by Tewodros Amdeberhan [Amd16] and proved by Straub [Str16].…”

Section: The Combinatorial Central Limit Theoremmentioning

confidence: 70%

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The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

Komlós

Sergel

Tusnády

2020

Electron. J. Combin.

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Simultaneous core partitions are important objects in algebraic combinatorics. Recently there has been interest in studying the distribution of sizes among all (s, t)-cores for coprime s and t. Zaleski (2017) gave strong evidence that when we restrict our attention to (s, s + 1)cores with distinct parts, the resulting distribution is approximately normal. We prove his conjecture by applying the Combinatorial Central Limit Theorem and mixing the resulting normal distributions.

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Section: The Combinatorial Central Limit Theoremmentioning

confidence: 70%

“…3 Normality for a fixed number of parts Armin Straub [Str16] gave the following beautiful characterization of our chosen objects: A partition λ into distinct parts is an (s, s + 1)-core if and only if it has perimeter ℓ(λ) + λ 1 − 1 ≤ s − 1.…”

Section: The Combinatorial Central Limit Theoremmentioning

confidence: 99%

The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

Komlós

Sergel

Tusnády

2020

Electron. J. Combin.

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“…Core partitions of numerous types of additional restrictions have long been studied, since they are closely related to the representation of symmetric group [15], the theory of cranks [13], Dyck-paths [1,3,28], and Euler's theorem [22]. To solve core problems, mathematicians provide many different tools, including t-abacus [3,15], Hasse diagram [27,28] and even ideas from quantum mechanics [16].…”

Section: Introductionmentioning

confidence: 99%

Bijections between t-core partitions and t-tuples

Zhong

2020

Discrete Mathematics

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This note introduces some bijections relating core partitions and tuples of integers. We apply these bijections to count the number of cores with various types of restriction, including fixed number of parts, limited size of parts, parts divisible by some integer, and distinct parts. For example, we prove that the number of 2t-core partitions into l even parts equals the number of t-core partitions into l parts. We also generalize one expression for simultaneous cores, which was given by Baek, Nam and Yu, recently. Subsequently, we use this expression to obtain recurrence satisfied by numbers of (s, s + 1, · · · , s + r)−core partitions for s ≥ 1.

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“…Zaleski [29] derived several explicit formulas for the k-th moment E[X k n,n+1 ] of X n,n+1 when k ≤ 16. The number, the largest size and the average size of (2n+1, 2n+3)-core partitions with distinct parts were also well studied (see [5,17,20,26,28]). Several explicit formulas for the k-th (when k ≤ 7) moment E[X k 2n+1,2n+3 ] of X 2n+1,2n+3 were obtained by Zaleski and Zeilberger [28].…”

Section: Introductionmentioning

confidence: 99%

“…In 2016, Straub [20] derived the following generalized Fibonacci recurrence for the number N d (n) of (n, dn − 1)-core partitions with distinct parts. Theorem 1.3 (Straub [20]).…”

Section: Introductionmentioning

confidence: 99%

On the polynomiality and asymptotics of moments of sizes for random (n, dn ± 1)-core partitions with distinct parts

Xiong

Zang

2019

Sci. China Math.

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Amdeberhan's conjectures on the enumeration, the average size, and the largest size of (n, n + 1)-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of (n, dn−1) and (n, dn+1)-core partitions with distinct parts, respectively. Let X s,t be the size of a uniform random (s, t)-core partition with distinct parts when s and t are coprime to each other. Some explicit formulas for the k-th moments] were given by Zaleski and Zeilberger when k is small. Zaleski also studied the expectation and higher moments of X n,dn−1 and conjectured some polynomiality properties concerning them in arXiv:1702.05634.Motivated by the above works, we derive several polynomiality results and asymptotic formulas for the k-th moments of X n,dn+1 and X n,dn−1 in this paper, by studying the beta sets of core partitions. In particular, we show that these k-th moments are asymptotically some polynomials of n with degrees at most 2k, when d is given and n tends to infinity. Moreover, when d = 1, we derive that the k-th moment E[X k n,n+1 ] of X n,n+1 is asymptotically equal to n 2 /10 k when n tends to infinity. The explicit formulas for the expectations E[X n,dn+1 ] and E[X n,dn−1 ] are also given. The (n, dn − 1)-core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of X n,dn−1 .

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Core partitions into distinct parts and an analog of Euler’s theorem

Cited by 39 publications

References 17 publications

The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

Bijections between t-core partitions and t-tuples

On the polynomiality and asymptotics of moments of sizes for random (n, dn ± 1)-core partitions with distinct parts

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